Function Space Optimization: A Symbolic Regression Method for Estimating Parameter Transfer Functions for Hydrological Models. Issue 10 (6th October 2020) - Feigl, M.; Herrnegger, M.; Klotz, D.; Schulz, K.
Abstract: Estimating parameters for distributed hydrological models is a challenging and long studied task. Parameter transfer functions, which define model parameters as functions of geophysical properties of a catchment, might improve the calibration procedure, increase process realism, and can enable prediction in ungauged areas. We present the function space optimization (FSO), a symbolic regression method for estimating parameter transfer functions for distributed hydrological models. FSO is based on the idea of transferring the search for mathematical expressions into a continuous vector space that can be used for optimization. This is accomplished by using a text generating neural network with a variational autoencoder architecture that can learn to compress the information of mathematical functions. To evaluate the performance of FSO, we conducted a case study using a parsimonious hydrological model and synthetic discharge data. The case study consisted of two FSO applications: single‐criteria FSO, where only discharge was used for optimization, and multicriteria FSO, where additional spatiotemporal observations of model states were used for transfer function estimation. The results show that FSO is able to estimate transfer functions correctly or approximate them sufficiently. We observed a reduced fit of the parameter density functions resulting from the inferred transfer functions for less sensitive model parameters. For those it was sufficient to estimate functions resulting in parameter distributions with approximately the same mean parameter values as the real transfer functions. The results of the multicriteria FSO showed that using multiple spatiotemporal observations for optimization increased the quality of estimation considerably. Plain Language Summary: Hydrological models are widely used tools for predicting river runoff or other components of the hydrological cycle that are important for the management of water resources. Typically, processes in those models use parameters to characterize the unique aspect of the studied area. Usually, these parameters are optimized to produce a well‐performing prediction model. This potentially leads to a loss of their physical meaning. Preserving the physical meaning of model parameters can be achieved by defining them with a relationship to properties of the modeled area (soil properties, topography, etc). These relationships are given as mathematical equations that compute parameters from a set of geophysical properties. We here present a method to automatically estimate such equations, called function space optimization (FSO). FSO transfers the search for mathematical equations into an optimization problem by using a neural network to encode the information of potential equations. We show FSO's ability in a case study using a hydrological model and synthetic runoff data. The results show that FSO is able to approximate the true relationship sufficiently. Furthermore, we show that additional spatial observation data can increase FSO performance. Key Points: This study introduces a method to infer transfer functions for the multiscale parameter regionalization approach We demonstrate its ability in a case study using synthetic runoff data We show that multicriteria optimization can improve the estimation of parameter transfer functions